In the manufacturing of electronic devices such as integrated circuits or magneto-resistive (MR) heads for use in data storage systems, a photo-lithography system is normally employed for transferring device features from a device mask to a wafer (or substrate) from which the devices are to be made. Generally, the performance of the manufactured devices is heavily dependent on minimizing the mask to wafer superposition errors during this photo-lithography process. It is therefore critical that any misalignments between the mask and the wafer being used for making the devices are accurately predicted and detected so that the wafer can be properly accepted or rejected in the final disposition of the wafer.
An example of photo-lithography systems is described in U.S. Pat. No. 4,803,712, issued to Kembo et al., for "X-Ray Exposure System." For reference purposes, the diagram of a prior art photo-lithography system shown in the U.S. Pat. No. 4,803,712 is reproduced here as a system 1 in FIG. 1 with certain non-essential elements omitted. The system 1 comprises a frame 2 placed on a floor 11, an X-ray source 3 mounted on the frame 2, a mask alignment apparatus 9 for aligning a mask 6 and a wafer 7, and a bed 10 supporting the mask alignment apparatus 9 thereon. The X-ray source 3 radiates X-rays from a fixed X-ray generating point on a rotary cathode 4 onto mask 6 to expose a device pattern on the wafer.
FIG. 2 shows an exemplary semiconductor wafer 7 from which the features of the electronic devices are fabricated using the system 1 of FIG. 1. The wafer 7 is divided into a large number of sliders (also referred to as dies) 15 by scribe lines 14. A slider 15 is a portion of the wafer 7 from which an electronic device will be made. A flat edge 13 of the wafer 7 defines the orientation of the wafer while it is being processed. A set of the sliders 15 is used in measuring the mask-wafer misalignment and referred to as test points 16.
In order to assess the misalignment errors, a specialized microelectronic instrument is typically employed for measuring the magnitude and direction of the errors of the test points 16. Such a measuring instrument may be a Quaestor Q2 semiconductor optical metrology system manufactured by the Bio-Rad Micromeasurements Inc. of Mountain View, Calif.
The error measurements for the wafer 7 are used in two important areas. First, the misalignments provide information about the sources of errors occurring in the operation of the mask alignment apparatus in the photo-lithography system. This information permits appropriate alignment actions to be taken for minimizing the mask-to-wafer misalignment. The errors that affect the whole wafer are commonly referred to as systematic errors and typically occur due to translation, rotation, magnification, or dimensional changes of the photo mask in relation to previous masking operations. The systematic errors also include lateral distortions of the wafer during manufacturing which cause random shifts of the device features. Typically, the systematic errors for each wafer must be estimated based on the measured misalignments for its test points 16.
FIGS. 3 and 4 illustrate different types of misalignment between the wafer 7 and mask 6, and their respective sources. FIG. 3 shows the relative positions of a mask plane 22 in which the mask 6 would be positioned (not shown), a wafer plane 20 in which a wafer 7 would be positioned (not shown), and a lens assembly 21 for projecting the device features from the mask 6 onto the wafer 7. FIG. 4 illustrates some typical misalignment errors between the mask 6 and the wafer 7, and the sources of these errors.
In addition to mask alignment, the measured misalignments are considered in the disposition of the wafer 7 in accordance with its predetermined tolerance specifications. For instance, the assessed misalignment of the wafer is compared to its tolerance limits and the wafer is accepted or rejected, depending on whether the misalignment is within the acceptable limits. Usually, the manufacturer of the electronic devices is committed to achieve a certain quality level for its products. To insure this quality, a variable acceptance sampling procedure is normally used during the fabrication of the devices wherein the wafer 7 is accepted or rejected based on the sampled data. Common acceptance sampling procedures are described, for example, by A. J. Duncan in "Quality Control and Industrial Statistics," Irwin, 1986.
Prior art wafer disposition methods typically assume that the misalignments of the test points 16 have a common normal distribution with the same mean and standard deviation. That is, d.sub.i .about.N(.mu., .sigma..sup.2) where d.sub.i is the misalignment of the i-th test point in a direction in the wafer plane 20, and the mean .mu. and standard deviation .sigma. are unknown parameters. Using a specialized measuring instrument such as the Quaestor Q2 referred to above, the misalignments for n number of test points 16 are first obtained. The average misalignment d and sample standard deviation s are then calculated based on the measured misalignments of the test points 16.
The wafer 7 is accepted if the conditions (d-ks&gt;L) and (d+ks&lt;U) are satisfied, where L and U are the lower and upper bounds, respectively, of the specified tolerance specifications. The constant k is calculated so that if the probability of defective for the wafer is (1-.beta.), then it will be accepted only 100 .alpha. % of the time. The value of k is equal to ##EQU1## where the constant t.sub..alpha., .sqroot.nZ.sbsb..beta. is the 100.alpha. percentile of the noncentral t-distribution having a noncentral parameter .sqroot.nZ.sub..beta. and a degree of freedom n-1, and Z.sub..beta. is the 100.beta. percentile for the standard normal distribution. .alpha. and .beta. are specified by the customer and usually take the values of 0.95 and 0.975, respectively.
A drawback of prior art wafer disposition methods based on normal distribution sampling is that they often assume the samples are randomly chosen from the same normal population and ignore the systematic errors. However, due to manufacturing constraints, the locations of the test points 16 generally must remain the same from one wafer to another. Unfortunately, the misalignment of any test point 16 is highly dependent on its location on the wafer 7. For example, the test points 16 near an edge 17 of the wafer 7 tend to have a higher rate of misalignment errors than those near its center. The test points 16 are thus not necessarily representative of the remaining sliders 15 on the wafer 7.
As an example, FIG. 5 illustrates a three-dimensional graph of a typical distribution of the misalignment errors of a wafer 7 used in the manufacturing of magneto-resistive heads for data storage systems. The graph of FIG. 6 shows in general that the sliders 15 near the wafer edge have a higher rate of misalignment errors than those closer to the center of the wafer. This error difference is illustrated, for instance, by a high point 30 and low point 31. The points 30 and 31 correspond respectively to a slider near the edge of the wafer and a slider near its center.
Since systematic errors are not included in prior art acceptance sampling methods, the resulting disposition of a wafer may not be accurate. That is, the underkill and overkill rates for the wafers 7 may be unnecessarily increased. For instance, if the test points 16 are located closer to the edge 17 of the wafer 7, the method will have a high overkill rate and reject more wafers than necessary. On the other hand, if the test points 16 are located in the center of the wafer 7, the method will result in a high escape rate and more bad wafers are accepted.
Another drawback of prior art wafer disposition techniques based on normal distribution sampling is that the samples do not accurately reflect the actual distribution of the misalignment errors of a typical wafer. This is due to the fact that the misalignment of a randomly chosen test point 16 is not normally distributed, but is actually a mixture of thousands of normal distributions. FIG. 6 illustrates the normal distributions for individual test points 16, while FIG. 7 shows the combined distribution of the misalignment errors for all sliders 15. Due to the inaccuracy of normal distribution sampling in this case, prior art wafer disposition methods based on such a sample may result in an inaccurate disposition of the wafers.
Moreover, in an actual manufacturing environment, it is desirable that a wafer disposition method must be sufficiently simple so that it can be can be easily implemented and incorporated into the manufacturing process.
In the paper "Normal Distribution Tolerance Limits for Stratified Random Samples," R. W. Mee addressed the problem of normal distribution sampling above by proposing methods for computing tolerance limits from a stratified random sample. However, the proposed methods cannot be applied to the present wafer disposition situation because they can only be used for wafers with a very small number of sliders, such as those having less than 10 sliders.
For a stratified random sampling, the sampled population is divided into nonoverlapping groups or strata. In a wafer disposition application, the strata would be the test points or sliders on the wafer. The proposed methods assumed that the characteristic of interest is normally distributed within each stratum and that the within-stratum variances are equal. Accordingly, the best approximate upper .beta.-content tolerance limit is given as: EQU (d.+-.Z.sub..beta. cs)+t.sub..alpha. (n-p)s1/n+c.sup.2 V!.sup.0.5( 1)
where d is the sample average, s is the standard deviation, n is the number of test points, p is the number of parameters in the model, and c is equal to (1+1/n).sup.0.5. The statistics Z.sub..beta. an estimator of the standardized percentile and must be iteratively calculated for each sample using a root-finding routine, while the value of V is an estimate for the variance Z.sub..beta. s/.sigma.. The number of terms in the expression (1) is proportional to the number of strata, which equals to n, the number of test points 16 on the wafer in this case.
Accordingly, a drawback of the proposed methods is that, because the calculation of the tolerance limits are very computationally intensive, they can not be generalized to a mixture of thousands of normal distributions for the randomly chosen wafer test points. The methods for computing tolerance limits described by R. W. Mee thus are not very practicable in an actual manufacturing environment. For a more detailed description on statistical tolerance limits, see, for instance, "Statistical Intervals, A Guide for Practitioners," G. J. Hahn and W. Q. Meeker, Wiley, 1991.
Still other prior art techniques studied the related problem of finding tolerance limits (or intervals) for random-effect model. See, for example, "Tolerance Intervals for the Distribution of True Values in the Presence of Measurement Errors," Technometrics, 36, No.2, pp.162-170, May 1994. However, these techniques donot provide a confidence limit for the combined probability of defective for all sliders in a wafer.
Therefore, there remains a need for a wafer disposition method that takes into account systematic errors in assessing the wafer-mask misalignment, provides an accurate disposition of the wafer with a confidence limit of the probability of defect for the wafer, and is sufficiently simple so that it can be easily implemented on-line in an actual manufacturing environment.